Finite Group Databasebeta

← #403435

Subgroups — #403435

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#2026C211yesmaximal, characteristic
4#14261C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11067C212yesmaximal
9#10932C212yesmaximal
10#10907C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16#2330—24no—
17#2330—24no—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×#376C2211yescharacteristic
28#2095—24no—
29#2026—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1443—22no—
40#1442—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#125—24no—
47C2×#125—24no—
48C2×#125—22no—
49C2×#125D414yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—24no—
65C2×#107—42no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—42no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391—42no—
76#391Q814yes—
77#389—216no—
78#389—24no—
79#389—24no—
80#389—42no—
81#387—216no—
82#384—22no—
83#384D414yes—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86#376—22no—
87#376—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#336—24no—
105#336—22no—
106#336—22no—
107#336—24no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8C2×Q812yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8#4114yes—
143#125—216no—
144#125—216no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4#4114yes—
179C2×C4⋊C4#4114yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—28no—
191C2×C42—82no—
192C2×C42—82no—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×Q812yes—
196C2×C42#4112yes—
197#107—48no—
198#103—48no—
199#103—48no—
200#95—28no—
201#95—28no—
202#95—24no—
203#95—24no—
204#95—28no—
205#95—42no—
206#95—42no—
207#95—42no—
208#95—44no—
209#95—82no—
210#95—82no—
211C24C22×D411yescharacteristic
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4#12014yes—
287C22×C4C22⋊Q814yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+411yescharacteristic
297C22×C42−1+411yescharacteristic
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—232no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—88no—
327C4⋊C4—88no—
328C4⋊C4—84no—
329C4⋊C4#12318yes—
330C42—24no—
331C42—24no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—42no—
337C42—42no—
338C42—88no—
339C23C2×#12011yescharacteristic
340C23C2×C22⋊Q811yescharacteristic
341C23#52511yescharacteristic
342C23#52511yescharacteristic
343C23#53211yescharacteristic
344C23#53511yescharacteristic
345C23#53611yescharacteristic
346C23#57612yes—
347C23#57712yes—
348C23#57812yes—
349C23#57812yes—
350Q8—44no—
351Q8—44no—
352Q8—82no—
353Q8—82no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4C2×#12312yes—
415C2×C4#52512yes—
416C2×C4#52912yes—
417C2×C4#53312yes—
418C2×C4#54212yes—
419C2×C4#54212yes—
420C2×C4#54612yes—
421C2×C4#54811yescharacteristic
422C2×C4#54811yescharacteristic
423C2×C4D4×Q811yescharacteristic
424C2×C4D4×Q811yescharacteristic
425C2×C4#55012yes—
426C2×C4#55212yes—
427C2×C4#55412yes—
428C2×C4#55612yes—
429C2×C4#55912yes—
430C8—44no—
431C8—82no—
432C22#239611yescharacteristic
433C22#240911yescharacteristic
434C22#243111yescharacteristic
435C22#258011yescharacteristic
436C22#260411yescharacteristic
437C22#263211yescharacteristic
438C22#284911yescharacteristic
439C22#301112yes—
440C22#301212yes—
441C22#308212yes—
442C22#308312yes—
443C22#312012yes—
444C22#312012yes—
445C22#312412yes—
446C22#312412yes—
447C22#317712yes—
448C22#317912yes—
449C22#322011yescharacteristic
450C22#322111yescharacteristic
451C22#322311yescharacteristic
452C22#322311yescharacteristic
453C22#324212yes—
454C22#324312yes—
455C4—44no—
456C4—44no—
457C4—48no—
458C4—81no—
459C4—81no—
460C4—84no—
461C4—82no—
462C4—82no—
463C4—161no—
464C4—161no—
465C4#242312yes—
466C4#244812yes—
467C4#245812yes—
468C4#246512yes—
469C2#1487511yescharacteristic
470C2#2379212yes—
471C2#2431112yes—
472C2#2560911yescharacteristic
473C2#2561111yescharacteristic
474C2#2926611yescharacteristic
475C2#2926811yescharacteristic
476C2#2953112yes—
477C2#3007211yescharacteristic
478C2#3007311yescharacteristic
479C2#3301712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23302
C2×#20952
C2×#20262
#142612
#113852
#113732
#113282
#110672
#109322
#109072
#107962